# Open Ball in Euclidean 3-Space is Interior of Sphere

## Theorem

Let $\R^3$ be the real Euclidean $3$-space considered as a metric space under the usual metric.

Let $x = \tuple {x_1, x_2, x_3} \in \R^3$ be a point in $\R^3$.

Let $\map {B_\epsilon} x$ be the open $\epsilon$-ball at $x$.

Then $\map {B_\epsilon} x$ is the interior of the sphere whose center is $x$ and whose radius is $\epsilon$.

## Proof

Let $S = \map {B_\epsilon} x$ be an open $\epsilon$-ball at $x$.

Let $y = \tuple {y_1, y_2, y_3} \in \map {B_\epsilon} x$.

Then:

 $\ds y$ $\in$ $\, \ds \map {B_\epsilon} x \,$ $\ds$ $\ds \leadstoandfrom \ \$ $\ds \map d {y, x}$ $<$ $\, \ds \epsilon \,$ $\ds$ Definition of Open $\epsilon$-Ball $\ds \leadstoandfrom \ \$ $\ds \sqrt {\paren {y_1 - x_1}^2 + \paren {y_2 - x_2}^2 + \paren {y_3 - x_3}^2}$ $<$ $\, \ds \epsilon \,$ $\ds$ Definition of Euclidean Metric $\ds \leadstoandfrom \ \$ $\ds \paren {y_1 - x_1}^2 + \paren {y_2 - x_2}^2 + \paren {y_3 - x_3}^2$ $<$ $\, \ds \epsilon^2 \,$ $\ds$
$\paren {y_1 - x_1}^2 + \paren {y_2 - x_2}^2 + \paren {y_3 - x_3}^2 = \epsilon^2$

is the equation of a sphere whose center is $x$ and whose radius is $\epsilon$.

The result follows by definition of interior and Open Ball of Point Inside Open Ball.

$\blacksquare$